4 Exericses 1. Verified that Silversteins sample covariance matrix theorem can be inferred directly from the Marcenko- Pastur theorem. Note: You will have to do some substitution tricks to get the parameter" c"to refer to the same quantity. 2. Derive the moments of the Wishart matrix from the observation that as c-0, the density becomes approximately" semi-circular. Hint: you will have to make an approximation for the region of suppor while remembering that for any a 1, a2< a. There will also be a shifting and rescaling in this problem to get the terms to match up correctly. ecall that the moments of the wishart matrix are expresse in terms of the Narayana polynomials in(34) 3. Come up with numerical code to compute the theoretical density when dH(T) has three atomic masses g. dH(=0.48(T-1)+0.48(T-3)+0.28(T-7). Plot the limiting density as a function of z for of values of 4. Do the same when there are four atomic masses in dh()(e. g. dH()=0.3 8(T-1)+0.25 8(T-3)+ 0.258(T-7)+0.25(T-10)). Verify that the solution obtained m rith the simulations Hints: Do all the roots match up 5. What happens if there are atomic masses of negative weight in dH(T)(e.g. dH()=0.58(7+1)+ 058(T-1). Does the limiting theoretical density line up with the experimental results? Check the ssumptions of the Marcenko-Pastur theorem! Is this" allowed"? References [1 Sang II Choi and Jack W. Silverstein. Analysis of the limiting spectral distribution of large dimensional andom matrices. volume 54(2), pages 295-309 1995 2 Boris A. Khoruzenkho, A M. Khorunzhy, and L.A. Pastur. Asymptotic properties of large random matrices with independent entries. Journal of Mathematical Physics, 37: 1-28, 1996 3 L Pastur. A simple approach to global regime of random matrix theory. In Mathematical results in statistical mechanics, Marseilles, 1998, pages 429-454. World Sci. Publishing, River Edge, NJ, 1999 4 Madan Lal Mehta. Random Matrices. Academic Press, Boston, second edition, 1991 5 M. Abramowitz and L.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications New York. 1970 6 N I. Akhiezer. The classical moment problem and some related questions in analysis. Hafner Publishing Co.. New York. New York. 1965. Translated by N. Kemmer [7] A M. Tulino and S. Verdi. Random matrices and wireless communications. Foundations and Trends in Communications and Information Theory, 1(1), June 2004 8 Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Math.,62:548-564,1955 9 V.A. Marcenko and L.A. Pastur. Distribution of eigenvalues for some sets of random matrices. Math USSR Sbornik. 1: 457-483. 1967. [10 Jack W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large dimensional andom matrices. Journal of Multivariate Analysis, 55(2): 331-339, 1995 11 Z D. Bai and J. W. Silverstein. On the empirical distribution of eigenvalues of a class of large dimensional random matices. Journal of Multivariate Analysis, 54(2): 175-192, 19954 Exericses 1. Verified that Silverstein’s sample covariance matrix theorem can be inferred directly from the Marˇcenko￾Pastur theorem. Note: You will have to do some substitution tricks to get the parameter “c” to refer to the same quantity. 2. Derive the moments of the Wishart matrix from the observation that as c → 0, the density becomes “approximately” semi-circular. Hint: you will have to make an approximation for the region of support while remembering that for any a < 1, a2 < a. There will also be a shifting and rescaling in this problem to get the terms to match up correctly. Recall that the moments of the Wishart matrix are expressed in terms of the Narayana polynomials in (34). 3. Come up with numerical code to compute the theoretical density when dH(τ) has three atomic masses (e.g. dH(τ) = 0.4 δ(τ − 1) + 0.4 δ(τ − 3) + 0.2 δ(τ − 7)). Plot the limiting density as a function of x for a range of values of c. 4. Do the same when there are four atomic masses in dH(τ) (e.g. dH(τ) = 0.3 δ(τ − 1) + 0.25 δ(τ − 3) + 0.25 δ(τ − 7) + 0.25 δ(τ − 10)) . Verify that the solution obtained matches up with the simulations. Hints: Do all the roots match up? 5. What happens if there are atomic masses of negative weight in dH(τ) (e.g. dH(τ) = 0.5 δ(τ + 1) + 0.5 δ(τ − 1)). Does the limiting theoretical density line up with the experimental results? Check the assumptions of the Marˇcenko-Pastur theorem! Is this “allowed”? References [1] Sang II Choi and Jack W. Silverstein. Analysis of the limiting spectral distribution of large dimensional random matrices. volume 54(2), pages 295–309. 1995. [2] Boris A. Khoruzenkho, A.M. Khorunzhy, and L.A. Pastur. Asymptotic properties of large random matrices with independent entries. Journal of Mathematical Physics, 37:1–28, 1996. [3] L. Pastur. A simple approach to global regime of random matrix theory. In Mathematical results in statistical mechanics, Marseilles, 1998, pages 429–454. World Sci. Publishing, River Edge, NJ, 1999. [4] Madan Lal Mehta. Random Matrices. Academic Press, Boston, second edition, 1991. [5] M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1970. [6] N. I. Akhiezer. The classical moment problem and some related questions in analysis. Hafner Publishing Co., New York, New York, 1965. Translated by N. Kemmer. [7] A. M. Tulino and S. Verd´u. Random matrices and wireless communications. Foundations and Trends in Communications and Information Theory, 1(1), June 2004. [8] Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Math., 62:548–564, 1955. [9] V.A. Marˇcenko and L.A. Pastur. Distribution of eigenvalues for some sets of random matrices. Math USSR Sbornik, 1:457–483, 1967. [10] Jack W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. Journal of Multivariate Analysis, 55(2):331–339, 1995. [11] Z. D. Bai and J. W. Silverstein. On the empirical distribution of eigenvalues of a class of large dimensional random matices. Journal of Multivariate Analysis, 54(2):175–192, 1995